Preheat the oven: Set to 350°F (175°C). Lightly butter a 2-quart baking dish.
Make the custard base: In a medium saucepan off the heat, whisk together 3/4 cup sugar, flour, cornstarch, and salt. Whisk in the milk gradually until smooth.
Cook the custard: Place the pan over medium heat and cook, whisking constantly, until the mixture thickens and starts to bubble, about 6–8 minutes.
Temper the yolks: In a small bowl, whisk the egg yolks.
Slowly whisk in 1/2 cup of the hot custard, then pour the yolk mixture back into the saucepan. Cook 1–2 more minutes, whisking, until thick and glossy. Remove from heat and stir in the butter and 1 1/2 teaspoons vanilla.
Layer the pudding: Arrange a layer of vanilla wafers on the bottom of the baking dish.
Add a layer of sliced bananas. Spoon about one-third of the warm custard over the bananas. Repeat layers (wafers, bananas, custard) two more times, finishing with custard on top.
Make sure all wafers are lightly covered.
Make the meringue: In a clean bowl, beat the egg whites and cream of tartar on medium-high until soft peaks form. Gradually add 1/4 cup sugar and 1/2 teaspoon vanilla, beating until glossy stiff peaks form.
Top and seal: Spread the meringue over the hot custard, making sure it touches the edges of the dish to seal. Use the back of a spoon to create peaks.
Bake: Bake for 12–15 minutes, until the meringue is golden brown.
Rotate the dish once if browning unevenly.
Cool and serve: Let the pudding cool 15–20 minutes before serving warm. For a chilled option, cool to room temperature, then refrigerate at least 3 hours.